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Ostwald ripening of droplets: The role of migration

Published online by Cambridge University Press:  01 February 2009

KARL GLASNER
Affiliation:
University of Arizona, 617 N. Santa Rita, Tucson, AZ 85721, U.S.A. email: kglasner@math.arizona.edu
FELIX OTTO
Affiliation:
University of Bonn, Wegelerstraβe 10, D–53115 Bonn, Germany email: otto@iam.uni-bonn.de; rump@iam.uni-bonn.de
TOBIAS RUMP
Affiliation:
University of Bonn, Wegelerstraβe 10, D–53115 Bonn, Germany email: otto@iam.uni-bonn.de; rump@iam.uni-bonn.de
DEJAN SLEPČEV
Affiliation:
Carnegie Mellon University, Pittsburgh, PA 15213-3890, U.S.A. email: slepcev@math.cmu.edu

Abstract

A configuration of near-equilibrium liquid droplets sitting on a precursor film which wets the entire substrate can coarsen in time by two different mechanisms: collapse or collision of droplets. The collapse mechanism, i.e., a larger droplet grows at the expense of a smaller one by mass exchange through the precursor film, is also known as Ostwald ripening. As was shown by K. B. Glasner and T. P. Witelski (‘Collision versus collapse of droplets in coarsening of dewetting thin films’, Phys. D209 (1–4), 2005, 80–104) in case of a one-dimensional substrate, the migration of droplets may interfere with Ostwald ripening: The configuration can coarsen by collision rather than by collapse. We study the role of migration in a two-dimensional substrate for a whole range of mobilities. We characterize the velocity of a single droplet immersed into an environment with constant flux field far away. This allows us to describe the dynamics of a droplet configuration on a two-dimensional substrate by a system of ODEs. In particular, we find by heuristic arguments that collision can be a relevant coarsening mechanism.

Type
Papers
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Alikakos, N. D., Bates, Peter W. & Xinfu, Chen (1994) Convergence of the Cahn–Hilliard equation to the Hele– Shaw model. Arch. Rational Mech. Anal. 128 (2), 165205.CrossRefGoogle Scholar
[2]Alikakos, N. D. & Fusco, G. (2003) Ostwald ripening for dilute systems under quasistationary dynamics. Comm. Math. Phys. 238 (3), 429479.Google Scholar
[3]Alikakos, N. D., Fusco, G. & Karali, G. (2003) The effect of the geometry of the particle distribution in Ostwald ripening. Comm. Math. Phys. 238 (3), 481488.CrossRefGoogle Scholar
[4]Alikakos, N. D., Fusco, G. & Karali, G. (2004) Ostwald ripening in two dimensions – the rigorous derivation of the equations from the Mullins–Sekerka dynamics. J. Differ. Eq. 205 (1), 149.CrossRefGoogle Scholar
[5]Constantin, P., Dupont, T. F., Goldstein, R. E., Kadanoff, L. P., Shelley, M. J. & Zhou, S.-M. (June 1993) Droplet breakup in a model of the Hele–Shaw cell. Phys. Rev. E 47 (6), 41694181.CrossRefGoogle Scholar
[6]Elliott, C. M. & Garcke, H. (1996) On the Cahn–Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27 (2), 404423.CrossRefGoogle Scholar
[7]Glasner, K. Ostwald ripening in thin film equations. submitted.Google Scholar
[8]Glasner, K. B. & Witelski, T. P. (2003) Coarsening dynamics of dewetting films. Phys. Rev. E 67 (1), 016302.Google ScholarPubMed
[9]Glasner, K. B. & Witelski, T. P. (2005) Collision versus collapse of droplets in coarsening of dewetting thin films. Phys. D 209 (1–4), 80104.Google Scholar
[10]Greenspan, H. P. (1978) On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125143.CrossRefGoogle Scholar
[11]Onsager, L. (1931) Reciprocal relations in irreversible processes, ii. Phys. Rev. 38, 2265.CrossRefGoogle Scholar
[12]Oron, A., Davis, S. H. & Bankof, S. G. (1997) Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931980.CrossRefGoogle Scholar
[13]Otto, F., Rump, T. & Slepčev, D. (2006) Coarsening rates for a droplet model: rigorous upper bounds. SIAM J. Math. Anal. 38 (2), 503529 (electronic).CrossRefGoogle Scholar
[14]Pego, R. L. (1989) Front migration in the nonlinear Cahn–Hilliard equation. Proc. R. Soc. Lond., Ser. A 422 (1863), 261278.Google Scholar
[15]Pismen, L. M. & Pomeau, Y. (2004) Mobility and interactions of weakly nonwetting droplets. Phys. Fluids 16 (7), 26042612.CrossRefGoogle Scholar
[16]Podgorski, T., Flesselles, J.-M. & Limat, L. (2001) Corners, cusps, and pearls in running drops. Phys. Rev. Lett. 87, 036102.CrossRefGoogle ScholarPubMed
[17]Seemann, R., Herminghaus, S., Neto, C., Schlagowski, S., Podzimek, D., Konrad, R., Mantz, H. & Jacobs, K. (2005) Dynamics and structure formation in thin polymer melt films. J. Phys.: Condens. Matter 17, S267S290.Google Scholar
[18]Thiele, U., Neuffer, K., Bestehorn, M., Pomeau, Y. & Velarde, M. (2001) Sliding drops in the diffuse interface model coupled to hydrodynamics. Phys. Rev. E. 64, 061601.CrossRefGoogle ScholarPubMed
[19]Voorhees, P. & Ratke, L. (2001) Growth and Coarsening. Springer, Berlin.Google Scholar